Joseph D Rabinoff

Journal Article Journal of Algebra · April 15, 2019 For a scheme X defined over the length 2 p-typical Witt vectors W2(k) of a characteristic p field, we introduce total p-differentials which interpolate between Frobenius-twisted differentials and Buium's p-differentials. They form a sheaf over the reductio ... Full text Cite

Non-Archimedean and tropical theta functions

Journal Article Mathematische Annalen · December 1, 2018 We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedea ... Full text Cite

Diophantine and tropical geometry, and uniformity of rational points on curves

Conference Proceedings of Symposia in Pure Mathematics · January 1, 2018 We describe recent work connecting combinatorics and tropical/ non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty–Coleman lies at ... Full text Cite

Tropical skeletons

Journal Article Annales de l'Institut Fourier · January 1, 2017 In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of ... Full text Cite

Optimal quotients of Jacobians with toric reduction and component groups

Journal Article Canadian Journal of Mathematics · December 1, 2016 Let J be a Jacobian variety with toric reduction over a local field K. Let J → E be an optimal quotient defined over K, where E is an elliptic curve. We give examples in which the functorially induced map φJ → φE on component groups of the Neron models is ... Full text Cite

Skeletons and tropicalizations

Journal Article Advances in Mathematics · May 14, 2016 Let K be a complete, algebraically closed non-archimedean field with ring of integers K∙ and let X be a K-variety. We associate to the data of a strictly semistable K∙-model X of X plus a suitable horizontal divisor H a skeleton S(X,H) in the analytificati ... Full text Cite

Nonarchimedean geometry, tropicalization, and metrics on curves

Journal Article Algebraic Geometry · January 1, 2016 We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity form ... Full text Cite

Uniform bounds for the number of rational points on curves of small mordell-weil rank

Journal Article Duke Mathematical Journal · January 1, 2016 Let X be a curve of genus g ≥ 2 over a number field F of degree d = [F : Q]. The conjectural existence of a uniform bound N (g, d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Cap ... Full text Cite

Lifting harmonic morphisms i: Metrized complexes and berkovich skeleta

Journal Article Research in Mathematical Sciences · December 1, 2015 Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a me ... Full text Cite

The skeleton of the jacobian, the jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

Journal Article International Mathematics Research Notices · January 1, 2015 Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let be its value group. Given a smooth, proper, connected K-curve X and a skeleton of the Berkovich analytification Xan, there are two natu ... Full text Cite

Lifting harmonic morphisms ii: Tropical curves and metrized complexes

Journal Article Algebra and Number Theory · January 1, 2015 We prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the nonvanishing of certain Hurwitz numbers, and we give ... Full text Cite

Higher-level canonical subgroups for p-divisible groups

Journal Article Journal of the Institute of Mathematics of Jussieu · April 1, 2012 Let R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti-Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗ - ... Full text Cite

Tropical analytic geometry, Newton polygons, and tropical intersections

Journal Article Advances in Mathematics · April 1, 2012 In this paper we use the connections between tropical algebraic geometry and rigid-analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: i ... Full text Cite

Hybrid grids and the Homing Robot

Journal Article Discrete Applied Mathematics · May 15, 2004 In their paper (Inform. Process. Lett. 77 (2001) 261), Wongngamnit and Angluin introduced a memory-efficient robot, called the Homing Robot, which localizes in an occupancy grid. We present a more general class of grids called hybrid grids, and establish t ... Full text Cite

Interactive Linear Algebra

Internet Publication Free online linear algebra textbook aimed at first-year engineering students. ... Link to item Cite

A rigid analytic proof that the Abel-Jacobi map extends to compact-type models

Journal Article Canadian Mathematical Bulletin Cite